The Senate Banking Committee on Tuesday advanced three of President Barack Obama’s nominees for positions on top jobs at the Federal Reserve.

On a voice vote, the committee sent the nominations of Stanley Fischer, Lael Brainard and Jerome Powell to the Senate floor. The full chamber is expected to confirm all three to join the Fed’s board of governors, but no date has been set for those votes. There is a real risk they won’t be confirmed by June, a senior Senate Democratic aide told the Wall Street Journal Monday, a development which would leave the seven-seat board with just three members for the first time since 1936, when Congress created the current board structure.

The votes came as the Fed’s policy committee began a two-day meeting at which it is expected to keep scaling back its bond-buying program and leave short-term rates unchanged near zero.

Mr. Fischer, who ran the Israeli central bank for eight years until last summer, is Mr. Obama’s pick to become Fed vice chairman. Ms. Brainard, who served as undersecretary for international affairs at the Treasury Department until last November, has been nominated for a seat on the Fed board. Mr. Powell has served on the Fed board since 2012 and is up for a new term.

All three are “extremely well-qualified nominees,” said Sen. Tim Johnson (D., S.D.), the committee’s chairman. “It is important we move quickly to have as many governors in place as possible,” he added.

Praise also came from Sen. Mike Crapo (R., Idaho), the panel’s top Republican. Mr. Crapo said Mr. Fischer, Ms. Brainard and Mr. Powell have “demonstrated their understanding of the challenges facing the Federal Reserve board in returning to normalized monetary policy.”

A note of caution, though, came from Sen. David Vitter (R., La.), who has repeatedly called for more people with community-banking experience at the Fed. Academic economists now dominate the central bank’s board, he said, and “I think the board is not adequately diverse in terms of background.” Mr. Vitter was the only senator Tuesday to formally record a “no” vote on the three nominees.

Sen. Elizabeth Warren (D., Mass.), voted for Mr. Fischer but wrote in a commentary for Politico Magazine that she supported him “reluctantly” due to his time at Citigroup Inc., where he was vice chairman from 2002 and 2005. Ms. Warren said a growing number of senior U.S. government officials have ties to the bank. “When former Citigroup officials land top positions in government and former senior government officials land top positions at Citigroup time and again, Americans have good reason to question whether the interests of the people – or the big banks – is paramount,” she wrote.

Ms. Warren raised the same issue during Mr. Fischer’s confirmation hearing in March. At the time, he described his Citigroup experience as “extremely valuable” for his role as a bank regulator in Israel. “Without that experience, I would have come to it largely with an academic background without ever having seen the inside of a bank or, furthermore, without ever having worked in the private sector,” Mr. Fischer said at the hearing.

Fed governor, Jeremy Stein, plans to step down from the board on May 28. If the Senate hasn’t confirmed Mr. Fischer and Ms. Brainard by then, the board would have just three governors, Chairwoman Janet Yellen, Daniel Tarullo and Mr. Powell, who can continue serving in his current Fed slot pending confirmation.

Such a situation is unlikely to affect Fed policy, such as decisions on how quickly to scale back its bond-buying program or on when to raise interest rates. But it would make life more difficult for the Fed officials and staff running the central bank’s day-to-day operations.

Among the reasons why the Senate might not vote on the nominees by June is the number of other items crowding the Senate’s schedule, including other nominations. According to the White House, there are 265 nominees awaiting Senate votes, including 51 for judicial branch slots and 214 for executive-branch positions, including 33 ambassadors.

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